A270479 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/Prime(k).

1, 2, 4, 10, 83, 24527, 568662303, 597257071145398433, 701406190820966232276524545293793251, 983324663657314079928895427833949144985650475980190987462191430555529010

Offset

1,2

Comments

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1)) + r(2)/(n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 1..13

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

sqrt(3) - 1 = 1/(2*1) + 1/(3*2) + 1/(5*4) + 1/(7*10) + ...

Mathematica

r[k_] := 1/Prime[k]; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993, A000040.

Sequence in context: A182238 A110073 A090256 * A126140 A326967 A223851

Adjacent sequences: A270476 A270477 A270478 * A270480 A270481 A270482

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved